Swell-induced surface instability of confined hydrogel layers on substrates

In response to external stimuli, polymeric hydrogels can change volume and shape dramatically. Experimental studies have observed a variety of instability patterns of hydrogels, due to swelling or shrinking, many of which have not been well understood. The present paper considers swell-induced surface instability of a hydrogel layer on a rigid substrate. Based on a recently developed theoretical framework for neutral polymeric gels, a linear perturbation analysis is performed to predict the critical condition for the onset of the surface instability. Using a nonlinear finite element method, numerical simulations are presented to show the swelling process, with the evolution of initial surface perturbations followed by the formation of crease-like surface patterns. In contrast to previously suggested critical conditions for surface creasing, the present study suggests a material-specific condition that predicts a range of critical swelling ratios from about 2.5 to 3.4 and quantitatively relates the critical condition to material properties of the hydrogel system. A stability diagram is constructed with two distinct regions for stable and unstable hydrogels depending on two dimensionless material parameters.     

Finite element implementation of poroelasticity theory for swelling dynamics of hydrogels

Hydrogel can swell to many times of its dry volume, resulting in large deformation which is vital for its function. The swelling process is regulated by many physical and chemical mechanisms, and can, to some extent, be fairly described by the poroelasticity theory. Implementation of the poroelasticity theory in the framework of finite element method would aid the design and optimization of hydrogel-based soft devices. Choosing chemical potential and displacement as two field variables, we present the implementation of poroelasticity tailored for hydrogel swelling dynamics, detail the normalization of physical parameters and the treatment of boundary conditions. Several examples are presented to demonstrate the feasibility and correctness of the proposed strategy.     

数值模拟离子强度敏感水凝胶的多场特性

发展了多物理模型来研究溶液中离子强度敏感水凝胶电-力-化学多场耦合的特性. 模 型的主要控制方程包括：计算水凝胶内外离子浓度分布的Nernst-Planck化学场方程；描述 膨胀变形的力学场方程和描述电场的泊松方程. 无网格有限云团法和牛顿迭代法用来数值离 散和求解控制方程. 通过对比多场耦合的响应, 包括胶的膨胀率和胶内外离子浓度和电势的分布，探讨了影响胶体变形的主要因素. 对数值模拟结果和实验结果进行了对照.     

两性pH敏感凝胶瞬态溶胀问题的基本方程和数值解

聚电解质两性凝胶的侧链有可电离的酸性和碱性基团,因其能对所处溶液环境的pH值产生响应,又可称作两性pH敏感凝胶．两性pH敏感凝胶属于超弹性材料,其变形问题涉及力学-化学耦合．同时,由于凝胶的变形伴随着溶剂分子的迁移,导致变形不能瞬间完成,故应研究两性pH敏感凝胶变形发展的全过程．由于其实用价值高,研究难度大,两性pH敏感凝胶的瞬态溶胀问题近年来引起了国内外众多研究人员的关注．本文以非线性连续介质力学理论和孔隙弹性理论为基础,结合电离平衡、唐南平衡等条件,获得了能描述两性pH敏感凝胶材料在外界激励下形变发展全过程的基本方程,探究了外加载荷、溶液pH值、盐溶液浓度等因素对两性pH敏感凝胶瞬态溶胀的影响,所得结果对该材料的应用有一定参考价值．     

An EBE finite element method for simulating nonlinear flows in rotating spheroidal cavities

Many planetary and astrophysical bodies are rotating rapidly, fluidic and, as a consequence of rapid rotation, in the shape of an ablate spheroid. We present an efficient element‐by‐element (EBE) finite element method for the numerical simulation of nonlinear flows in rotating incompressible fluids that are confined in an ablate spheroidal cavity with arbitrary eccentricity. Our focus is placed on temporal and spatial tetrahedral discretization of the EBE finite element method in spheroidal geometry, the EBE parallelization scheme and the validation of the nonlinear spheroidal code via both the constructed exact nonlinear solution and the special resonant forcing in the inviscid limit. Copyright © 2009 John Wiley & Sons, Ltd.     

Flow simulation on moving boundary‐fitted grids and application to fluid–structure interaction problems

We present a method for the parallel numerical simulation of transient three‐dimensional fluid–structure interaction problems. Here, we consider the interaction of incompressible flow in the fluid domain and linear elastic deformation in the solid domain. The coupled problem is tackled by an approach based on the classical alternating Schwarz method with non‐overlapping subdomains, the subproblems are solved alternatingly and the coupling conditions are realized via the exchange of boundary conditions. The elasticity problem is solved by a standard linear finite element method. A main issue is that the flow solver has to be able to handle time‐dependent domains. To this end, we present a technique to solve the incompressible Navier–Stokes equation in three‐dimensional domains with moving boundaries. This numerical method is a generalization of a finite volume discretization using curvilinear coordinates to time‐dependent coordinate transformations. It corresponds to a discretization of the arbitrary Lagrangian–Eulerian formulation of the Navier–Stokes equations. Here the grid velocity is treated in such a way that the so‐called Geometric Conservation Law is implicitly satisfied. Altogether, our approach results in a scheme which is an extension of the well‐known MAC‐method to a staggered mesh in moving boundary‐fitted coordinates which uses grid‐dependent velocity components as the primary variables. To validate our method, we present some numerical results which show that second‐order convergence in space is obtained on moving grids. Finally, we give the results of a fully coupled fluid–structure interaction problem. It turns out that already a simple explicit coupling with one iteration of the Schwarz method, i.e. one solution of the fluid problem and one solution of the elasticity problem per time step, yields a convergent, simple, yet efficient overall method for fluid–structure interaction problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Transient analysis of temperature-sensitive neutral hydrogels

A model for transient deformation of neutral hydrogels that takes into account conservation of momentum, energy and mass for the solid polymer and fluid phase is derived, nondimensionalized and analyzed. Slow- and fast-response hydrogels are studied for three cases based on the response of (i) a spherical hydrogel, (ii) a constrained hydrogel slab to a step change in temperature, and (iii) the deformation in a temperature gradient. Model predictions for case (i) are shown to agree well with experiments for swelling and shrinking. For case (ii), solvent can be seen entering at the sides and flowing into the interior and towards the corners, such that the corners undergo a faster deformation than the sides. Immersed in a temperature gradient, case (iii), the hydrogel undergoes a bending motion and reaches a curved equilibrium shape, similar to the bending motion of polyelectrolyte hydrogels subjected to an external electric field. The benefit of the scale analysis conducted here, to predict correctly, prior to numerical computations, important characteristics such as stress, osmotic pressure and deformation times, is also highlighted.     

Fully coupled flow-induced vibration of structures under small deformation with GMRES method

Lagrangian-Eulerian formulations based on a generalized variational principle of fluid-solid coupling dynamics are established to describe flow-induced vibration of a structure under small deformation in an incompressible viscous fluid flow. The spatial discretization of the formulations is based on the multi-linear interpolating functions by using the finite element method for both the fluid and solid structures. The generalized trapezoidal rule is used to obtain apparently non-symmetric linear equations in an incremental form for the variables of the flow and vibration. The nonlinear convective term and time factors are contained in the non-symmetric coefficient matrix of the equations. The generalized minimum residual （GMRES） method is used to solve the incremental equations. A new stable algorithm of GMRES-Hughes-Newmark is developed to deal with the flow-induced vibration with dynamical fluid-structure interaction in complex geometries. Good agreement between the simulations and laboratory measurements of the pressure and blade vibration accelerations in a hydro turbine passage was obtained, indicating that the GiViRES-Hughes-Newmark algorithm presented in this paper is suitable for dealing with the flow-induced vibration of structures under small deformation.     

Implicit symmetrized streamfunction formulations of magnetohydrodynamics

We apply the finite element method to the classic tilt instability problem of two‐dimensional, incompressible magnetohydrodynamics, using a streamfunction approach to enforce the divergence‐free conditions on the magnetic and velocity fields. We compare two formulations of the governing equations, the standard one based on streamfunctions and a hybrid formulation with velocities and magnetic field components. We use a finite element discretization on unstructured meshes and an implicit time discretization scheme. We use the PETSc library with index sets for parallelization. To solve the nonlinear problems on each time step, we compare two nonlinear Gauss‐Seidel‐type methods and Newton's method with several time‐step sizes. We use GMRES in PETSc with multigrid preconditioning to solve the linear subproblems within the nonlinear solvers. We also study the scalability of this simulation on a cluster. Copyright © 2008 John Wiley & Sons, Ltd.     

Finite Elastic Deformations in Liquid-Saturated and Empty Porous Solids

Based on the Theory of Porous Media (TPM), a formulation of a fluid-saturated porous solid is presented where both constituents, the solid and the fluid, are assumed to be materially incompressible. Therefore, the so-called point of compaction exists. This deformation state is reached when all pores are closed and any further volume compression is impossible due to the incompressibility constraint of the solid skeleton material. To describe this effect, a new finite elasticity law is developed on the basis of a hyperelastic strain energy function, thus governing the constraint of material incompressibility for the solid material. Furthermore, a power function to describe deformation dependent permeability effects is introduced.After the spatial discretization of the governing field equations within the finite element method, a differential algebraic system in time arises due to the incompressibility constraint of both constituents. For the efficient numerical treatment of the strongly coupled nonlinear solid-fluid problem, a consistent linearization of the weak forms of the governing equations with respect to the unknowns must be used.     

An operator splitting numerical scheme for thermal/isothermal incompressible viscous flows

A numerical scheme for time‐dependent incompressible viscous fluid flow, thermally coupled under the Boussinesq approximation is presented. The scheme combines an operator splitting in the time discretization and linear finite elements in the space discretization, and is an extension of one previously applied for isothermal incompressible viscous flow governed by the Navier–Stokes equations. To show the efficiency of the scheme, numerical results are presented for mixed convection, and natural convection at high Rayleigh numbers. Restricting the scheme to the isothermal case, some numerical results at high Reynolds numbers are included, i.e. the scheme is tested for a small viscosity and a large force term, which are not trivial tasks to deal with. Copyright © 1999 John Wiley & Sons, Ltd.     

Analysis of shear-thinning fluid flow in intermeshing twin-screw extruders

Summary The paper deals with the analysis of three-dimensional incompressible flows of generalized Newtonian liquids in corotating twin-screw equipment where the fluid-filled region is rather complex and changes in time as the screws rotate. Taking notice of the fact that the flow fields are periodic in time and, simultaneously, periodic in axial direction, a mixed finite element method is applied for the sake of discretization. The resulting systems of nonlinear algebraic equations are treated by a Newton iteration where the linear subproblems with indefinite sparse coefficient matrices are solved by applying a modified Uzawa algorithm. By simulating different working conditions, the screw characteristics are extracted numerically, and the influence of Deborah number is demonstrated. Selected flow fields are visualized grafically illustrating the intermeshing screw operation principle and demonstrating certain symmetries connected with the kinematic reversibility of nonlinear viscous creeping flows. Accepted for publication 7 July 1996     

二维定常不可压缩粘性流动N-S方程的数值流形方法

将流形方法应用于定常不可压缩粘性流动N-S方程的直接数值求解,建立基于Galerkin加权余量法的N-S方程数值流形格式,有限覆盖系统采用混合覆盖形式,即速度分量取1阶和压力取0阶多项式覆盖函数,非线性流形方程组采用直接线性化交替迭代方法和Nowton-Raphson迭代方法进行求解.将混合覆盖的四节点矩形流形单元用于阶梯流和方腔驱动流动的数值算例,以较少单元获得的数值解与经典数值解十分吻合.数值实验证明,流形方法是求解定常不可压缩粘性流动N-S方程有效的高精度数值方法.     

Iterative solution techniques for the stokes and Navier-Stokes equations

The linear system arising from a Lagrange-Galerkin mixed finite element approximation of the Navier–Stokes and continuity equations is symmetric indefinite and has the same block structure as a system arising from a mixed finite element discretization of a Stokes problem. This paper considers the iterative solution of such a system, comparing the performance of the one-level preconditioned conjugate residual method for indefinite matrices with that of a more traditional two-level pressure correction approach. Asymptotic estimates for the amount of work involved in each method are given together with the results of related numerical experiments.     

Computational Aspects of Pseudospectra in Hydrodynamic Stability Analysis

This paper addresses the analysis of spectrum and pseudospectrum of the linearized Navier–Stokes operator from the numerical point of view. The pseudospectrum plays a crucial role in linear hydrodynamic stability theory and is closely related to the non-normality of the underlying differential operator and the matrices resulting from its discretization. This concept offers an explanation for experimentally observed instability in situations when eigenvalue-based linear stability analysis would predict stability. Hence the reliable numerical computation of the pseudospectrum is of practical importance particularly in situations when the stationary “base flow” is not analytically but only computationally given. The proposed algorithm is based on a finite element discretization of the continuous eigenvalue problem and uses an Arnoldi-type method involving a multigrid component. Its performance is investigated theoretically as well as practically at several two-dimensional test examples such as the linearized Burgers equations and various problems governed by the Navier–Stokes equations for incompressible flow.     

GURTIN VARIATIONAL PRINCIPLE AND FINITE ELEMENT SIMULATION FOR DYNAMICAL PROBLEMS OF FLUID-SATURATED POROUS MEDIA

Based on the theory of porous media,a general Gurtin variational principle for theinitial boundary value problem of dynamical response of fluid-saturated elastic porous media isdeveloped by assuming infinitesimal deformation and incompressible constituents of the solid andfluid phase.The finite element formulation based on this variational principle is also derived.Asthe functional of the variational principle is a spatial integral of the convolution formulation,thegeneral finite element discretization in space results in symmetrical differential-integral equationsin the time domain.In some situations,the differential-integral equations can be reduced to sym-metrical differential equations and,as a numerical example,it is employed to analyze the reflectionof one-dimensional longitudinal wave in a fluid-saturated porous solid.The numerical results canprovide further understanding of the wave propagation in porous media.     

潜堤上波流传播的完全非线性数值模拟

利用时域高阶边界元方法建立模拟波流混合作用的完全非线性数值水槽模型, 其中自由水面满足完全非线性自由水面条件. 采用混合欧拉-拉格朗日方法追踪流体瞬时水面, 运用4阶Runge-Kutta方法更新下一时间步的波面和速度势. 为了减少计算域, 提高计算速度, 采用同时消除底面和侧面的镜像格林函数; 在每一时间步内, 对网格进行重新划分以避免由于网格运动变形而引起的数值不稳定问题. 对水流中淹没潜堤上的波浪变形在水槽中开展了物理模型试验, 并把试验结果和数值结果进行了对比, 吻合得很好. 进一步研究了水流及潜堤的存在对高阶谐波产生的影响.      

Nonlinear axisymmetric deformations of an elastic tube under external pressure

The problem of the finite axisymmetric deformation of a thick-walled circular cylindrical elastic tube subject to pressure on its external lateral boundaries and zero displacement on its ends is formulated for an incompressible isotropic neo-Hookean material. The formulation is fully nonlinear and can accommodate large strains and large displacements. The governing system of nonlinear partial differential equations is derived and then solved numerically using the C++ based object-oriented finite element library Libmesh. The weighted residual-Galerkin method and the Newton-Krylov nonlinear solver are adopted for solving the governing equations. Since the nonlinear problem is highly sensitive to small changes in the numerical scheme, convergence was obtained only when the analytical Jacobian matrix was used. A Lagrangian mesh is used to discretize the governing partial differential equations. Results are presented for different parameters, such as wall thickness and aspect ratio, and comparison is made with the corresponding linear elasticity formulation of the problem, the results of which agree with those of the nonlinear formulation only for small external pressure. Not surprisingly, the nonlinear results depart significantly from the linear ones for larger values of the pressure and when the strains in the tube wall become large. Typical nonlinear characteristics exhibited are the “corner bulging” of short tubes, and multiple modes of deformation for longer tubes.     

A new stable space–time formulation for two‐dimensional and three‐dimensional incompressible viscous flow

A space–time finite element method for the incompressible Navier–Stokes equations in a bounded domain in ?^d (with d=2 or 3) is presented. The method is based on the time‐discontinuous Galerkin method with the use of simplex‐type meshes together with the requirement that the space–time finite element discretization for the velocity and the pressure satisfy the inf–sup stability condition of Brezzi and Babu?ka. The finite element discretization for the pressure consists of piecewise linear functions, while piecewise linear functions enriched with a bubble function are used for the velocity. The stability proof and numerical results for some two‐dimensional problems are presented. Copyright © 2001 John Wiley & Sons, Ltd.     

Finite Element Analysis for Flow Around a Rotating Body using Chimera Method

In this paper, the Chimera method with the Schwarz algorithm, which is one of overlapping domain decomposition methods, is applied for a flow around a rotating body. The incompressible Navier–Stokes equations expressed in a non-inertial frame of reference are used for the governing equations. The implicit scheme with accuracy of the second order is used for the temporal discretization. The mixed finite element formulation with the iso-P2 P1/P1 elements for velocity and pressure elements is used for the spatial discretization. For numerical examples, two-dimensional analyses of flow around a circular cylinder and an ellipse cylinder which rotate uniformly in a uniform flow were performed, the validity of the present technique was verified and the characteristics of the flow were considered.